Self-assembly of 4-sided fractals in the Two-Handed Tile Assembly Model View Full Text


Ontology type: schema:ScholarlyArticle      Open Access: True


Article Info

DATE

2019-03

AUTHORS

Jacob Hendricks, Joseph Opseth

ABSTRACT

We consider the self-assembly of fractals in one of the most well-studied models of tile based self-assembling systems known as the Two-Handed Tile Assembly Model (2HAM). In particular, we focus our attention on a class of fractals called discrete self-similar fractals (a class of fractals that includes the discrete Sierpiński carpet). We present a 2HAM system that finitely self-assembles the discrete Sierpiński carpet with scale factor 1. Moreover, the 2HAM system that we give lends itself to being generalized and we describe how this system can be modified to obtain a 2HAM system that finitely self-assembles one of any fractal from an infinite set of fractals which we call 4-sided fractals. The 2HAM systems we give in this paper are the first examples of systems that finitely self-assemble discrete self-similar fractals at scale factor 1 in a purely growth model of self-assembly. Finally, we show that there exists a 3-sided fractal (which is not a tree fractal) that cannot be finitely self-assembled by any 2HAM system. More... »

PAGES

75-92

References to SciGraph publications

  • 2012-06. Reducing Tile Complexity for the Self-assembly of Scaled Shapes Through Temperature Programming in ALGORITHMICA
  • 2014. One Tile to Rule Them All: Simulating Any Tile Assembly System with a Single Universal Tile in AUTOMATA, LANGUAGES, AND PROGRAMMING
  • 2010-03. Self-assembly of discrete self-similar fractals in NATURAL COMPUTING
  • 2012-10. Approximate Self-Assembly of the Sierpinski Triangle in THEORY OF COMPUTING SYSTEMS
  • 2011. Exact Shapes and Turing Universality at Temperature 1 with a Single Negative Glue in DNA COMPUTING AND MOLECULAR PROGRAMMING
  • 2016-09. Strict Self-Assembly of Fractals Using Multiple Hands in ALGORITHMICA
  • 2016-02. The Two-Handed Tile Assembly Model is not Intrinsically Universal in ALGORITHMICA
  • 2014. Scaled Tree Fractals Do not Strictly Self-assemble in UNCONVENTIONAL COMPUTATION AND NATURAL COMPUTATION
  • 2009. Self-assembly of the Discrete Sierpinski Carpet and Related Fractals in DNA COMPUTING AND MOLECULAR PROGRAMMING
  • 2013-10. Self-Assembling Rulers for Approximating Generalized Sierpinski Carpets in ALGORITHMICA
  • 2009. Limitations of Self-assembly at Temperature One in DNA COMPUTING AND MOLECULAR PROGRAMMING
  • 2018-09-07. Hierarchical Growth Is Necessary and (Sometimes) Sufficient to Self-assemble Discrete Self-similar Fractals in DNA COMPUTING AND MOLECULAR PROGRAMMING
  • 2014-06. An introduction to tile-based self-assembly and a survey of recent results in NATURAL COMPUTING
  • 2017-04-28. Self-assembly of Shapes at Constant Scale Using Repulsive Forces in UNCONVENTIONAL COMPUTATION AND NATURAL COMPUTATION
  • 2017-06. Reflections on tiles (in self-assembly) in NATURAL COMPUTING
  • Identifiers

    URI

    http://scigraph.springernature.com/pub.10.1007/s11047-018-9718-6

    DOI

    http://dx.doi.org/10.1007/s11047-018-9718-6

    DIMENSIONS

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    40 schema:description We consider the self-assembly of fractals in one of the most well-studied models of tile based self-assembling systems known as the Two-Handed Tile Assembly Model (2HAM). In particular, we focus our attention on a class of fractals called discrete self-similar fractals (a class of fractals that includes the discrete Sierpiński carpet). We present a 2HAM system that finitely self-assembles the discrete Sierpiński carpet with scale factor 1. Moreover, the 2HAM system that we give lends itself to being generalized and we describe how this system can be modified to obtain a 2HAM system that finitely self-assembles one of any fractal from an infinite set of fractals which we call 4-sided fractals. The 2HAM systems we give in this paper are the first examples of systems that finitely self-assemble discrete self-similar fractals at scale factor 1 in a purely growth model of self-assembly. Finally, we show that there exists a 3-sided fractal (which is not a tree fractal) that cannot be finitely self-assembled by any 2HAM system.
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